On the dependent property of solutions for general higher order periodic differential equation

In this paper, the property of linearly dependence of solutions f ( z ) and f ( z + 2 π i ) for higher order linear differential equation f ( k ) ( z ) + A k − 2 ( z ) f ( k − 2 ) ( z ) + ⋯ + A 0 ( z ) f ( z ) = 0 , where A j ( z ) ( j = 0 , … , k − 2 ) are entire periodic coefficients and A s ( z ) ( s ∈ { 2 , … , k − 2 } ) is the dominant coefficient, is investigated.